The Annals of Probability

Large Deviations, Moderate Deviations and LIL for Empirical Processes

Liming Wu

Full-text: Open access

Abstract

Let $(X_n)_{n\geq 1}$ be a sequence of i.i.d. r.v.'s with values in a measurable space $(E, \mathscr{E})$ of law $\mu$, and consider the empirical process $L_n(f) = (1/n)\sum^n_{k=1} f(X_k)$ with $f$ varying in a class of bounded functions $\mathscr{F}$. Using a recent isoperimetric inequality of Talagrand, we obtain the necessary and sufficient conditions for the large deviation estimations, the moderate deviation estimations and the LIL of $L_n(\cdot)$ in the Banach space of bounded functionals $\mathscr{l}_\infty(\mathscr{F})$. The extension to the unbounded functionals is also discussed.

Article information

Source
Ann. Probab., Volume 22, Number 1 (1994), 17-27.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176988846

Digital Object Identifier
doi:10.1214/aop/1176988846

Mathematical Reviews number (MathSciNet)
MR1258864

Zentralblatt MATH identifier
0793.60032

JSTOR
links.jstor.org

Subjects
Primary: 60F10: Large deviations
Secondary: 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case) 60G50: Sums of independent random variables; random walks

Keywords
Large deviations moderate deviations law of iterated logarithm (LIL) isoperimetric inequality Smirnov-Kolmogorov theorem

Citation

Wu, Liming. Large Deviations, Moderate Deviations and LIL for Empirical Processes. Ann. Probab. 22 (1994), no. 1, 17--27. doi:10.1214/aop/1176988846. https://projecteuclid.org/euclid.aop/1176988846


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